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Full text of "On the representation of the stability region in oscillation problems with the aid of the Hurwitz determinants"

AiAC|\Tm-l34g' 





NATIONAL ADVISORY COMMITTEE 
FOR AERONAUTICS 

TECHNICAL MEMORANDUM 1348 



ON THE REPRESENTATION OF THE STABILITY REGION IN 

OSCILLATION PROBLEMS WITH THE AID OF THE 

HURWrrZ DETERMINANTS 

By E. Sponder 



Translation of *Zur Darstellung des Stabilitatsgebietes bei 

Schwingungsaufgaben mit Hilfe der Hurwitz- 

Determinanten." Schweizer Archiv, 

March 1950. 




Washington 
August 1952 



^ b ^ fl ( 00 ;J'/ 'I 



NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 



TECHNICAL MEMORANDUM 13^ 



ON THE REPRESENTATION OF THE STABILITY REGION IN 
OSCILLATION PROBLEMS WITH THE AID OF THE 
HURWITZ DETERMINANTS* 
By E. Sponder 

For oscillation phenomena which may also have an unstable course, 
it is customary to represent the regions where stability or instability 
prevails in a plane as functions of two parameters x and y. 

In order to determine whether stability exists at any point of the 
plane represented (thus, whether a disturbance of the oscillation phe- 
nomenon considered is damped in its course) it must be investigated 
whether all roots X. of the frequency (characteristic) equation 

}J^ + a-j^X^"-"- + . . . + aj^ = 



have a negative real part at this arbitrary point. The mathematical 
condition for this is known to be that the n Hurwitz determinants D^ 
to Dn which are formed from the coefficients aj^ to a^^ and are 

fimctions of the two parameters x and y mentioned before are all 
positive for this point. Resulting from this criterion and completely 
equivalent to it is the fact that the coefficients a-, to a^^ and only 

a few certain Hurwitz determinants must be positive. 

If, conversely, all roots \ of the frequency equation have a nega- 
tive real part, all values D^ to T)-^ are positive. If one now visu- 
alizes the point considered before (for which we assume stability to have 
been established) as traveling in the representation plane of figure 1, 
there vary with its parameters -x and y also the coefficients a-^ 

to a-^y, the real parts of the roots X, and finally the n Hurwitz- 
determinants. If one arrives at a point where for instance a real root 

*"Ziar Darstellung des Stabilitatsgebietes bei Schwingungsaufgaben 
mit Hilfe der Hurwitz- Determinanten." Schweizer Archiv, March 1950, 
pp. 93-96. 



2 NACA TM 13^ 

disappears, a^j also disappears since |aj^ is the product of the values 

of all roots; if one reaches, in contrast, a point where the real part 
of a complex root becomes zero, it can be shown that then Dn_l becomes 
zero. For every case, however, the product ^-dP-^-^ disappears which is 
nothing else but the Hurwitz determinant of the nth degree 



Sn^n-l = ^n 



Thus the important theorem, the proof for which will be presented 
later, is valid: 

The limits of the (usually unique) stable region lie at 
^n = a^Dn.! =0. 

For graphical representation, it is therefore completely sufficient 
to plot Dj^ = or more simply ^n ~ '^ ^^^ ■'^n-1 ~ "^ ^^ separate 
limiting curves oX a region for which stability is known to prevail at 
an arbitrary point, as illustrated by figure 1. If the limit Eji-l ~ ^ 
is exceeded, the course of the oscillation process is "dynamically" 
unstable because a damping becomes negative; beyond the limit a^^ = 0, 
one usually calls the' oscillation process "statically" unstable. 

In particular, the following is valid for oscillation phenomena 
which lead to frequency equations of the 4th degree (reference l). 

The Hurwitz determinant of the 4th degree formed from the coeffi- 
cients A to E of the frequency equation 



AX^ + BX.3 + CX^ + BX + E = 



reads 



2 T,2^ 



D^ = ED3 = E(BCD - AD'^ - B'^E) = ER 



with the expression in parentheses known as Routh's discriminant R; the 
latter is therefore nothing else but the Hurwitz determinant of the 



NACA TM 13^3 



3rd degree. Since the coefficient A is usually +1, it is generally 
valid as dynamic stability condition that the expression (BC - D)D - B% 
turns out positive. The static stability is then guaranteed in addition 
by E > so that the graphic representation of the region of figure 2 
results. 

Therewith, every requirement has been met; for it is impossible 
that within the region denoted as stable a curve C = or D = 
could r\in and perhaps still further reduce this region. 

It is therefore completely superfluous to investigate further what 
sign the other coefficients of the frequency equation have once the 
boundaries for a region E = and R = have been fixed; however, 
and this is important, it must be known that, at an arbitrary point of 
this region, stability actually prevails. For this it is sufficient 
to check for a point conveniently situated (for instance on an axis) 
the signs of the remaining coefficients A to D of the frequency 
equation which is equivalent with the fact that there all Hurwitz deter- 
minants Di to 'Dl^. turn out positive. That this is important can, for 
instance, be recognized from the fact that, in the region indicated on 
the left in the plane of representation of figure 2, instability may 
prevail in spite of E > and R > 0. 

The advantage of these recognized facts given above does not result 
in much simplification for frequency equations of the 4th, 5th, or even 
6th degree; however, this advantage may save a great deal of unnecessary 
calculations. The expressions for the Hurwitz determinants become more 
and more cumbersome for higher degrees so that it will probably be a 
very welcome facilitation for the outlining of the stability region if 
the mere representation of the two curves a^ = and Dn-1 = will 
be sufficient. 

There now follows the proof of the theorem given before that the 
limits of the (usually imique) stable region lie at D^ = ^n-'^n-l ~ ^' 

1. For the present consideration, the existence of at least one 
stable region is presupposed. Not even one need necessarily always 
exist; on the other hand, s^eral stability regions, separated from 
one another, may occur. One can easily make sure of this by considering 
a frequency equation of the 2nd degree which pertains to an ordinary 
oscillation 

\ 

p 

X, + a-.X + 32 = 



MCA TM I3W 



The coefficients a^^ and 82 are dependent on two parameters x 
and y, corresponding to the two coordinate axes of the plane of repre- 
sentation. For simplicity's sake, we shall assume that a2 everywhere 
has positive values only; then the sign of a-^ alone is the criterion 
for the stability, since a^ is a measvire of the damping of the oscil- 
lation considered and this oscillation is stable only if a-, is posi- 
tive. The limit of stability now depends entirely on the sign of the 
analytical (or empirical) function of the coefficient a-, of the two 



parameters x and y, or, in other words, what form the curves 



= 



have in the plane of representation which separate the positive from the 
negative values. In this manner, one can easily understand that several 
stability regions may exist but that just as well not even one need exist 
anywhere in the entire plane of representation. 

For frequency equations of higher degree, the probability decreases 
that simultaneously in several regions all stability conditions found by 
Hurwitz will be satisfied. Usually, one will deal with only one single 
stability region which will then be considered more thoroughly. 

2. Hiorwitz' criterion (reference 2) signifies that a prescribed 
equation of the nth degree with real coefficients 



iH 



>n-l 



ar,\ + anX,""-^ + . . . + a-„ = 



(ao >0) 



possesses roots exclusively with negative real parts only when the values 
of the n-determinants 



% 



ai a3 a^ 

ao ag a^ 
an a-) 



^2k- 


-1 


32k- 


-2 


32k- 


■3 


• 




3k 





(k = 1,2,. ..,n) 

are all positive. Therein, one generally has to put a^ = when the 
subscript x is negative or larger than n. 



NACA TM 13^ 5 

It shall now be proved that the Hiirwitz determinant of the (n - l)th 
degree Dr,.! always disappears when the real part of a complex pair of 

roots is zero. 

Following the way Hurvitz piursued in deriving his criterion, one 
finds the remarkable presupposition that no piorely imaginary roots must 
exist. This, however, is precisely the condition whose influence on a 
certain Hiirwitz -determinant is of special interest. However, for that 
reason, one need not abandon this presupposition; it is sufficient to 
interpret Ej^_i simply as a prescribed calculation rule which one applies, 

without consideration of its derivation, to the coefficients of an equa- 
tion of the nth degree. 

It will now be expedient to consider (corresponding to the assump- 
tion that the real part of a complex pair of roots is to be zero) the 
equation of the nth degree as split into two factors: one is a quad- 
ratic expression with the purely imaginary pair of roots and the other 
is an expression of the (n - 2)th degree, thus 



[x^ + a)(ao>^'^"^ + ajX^'^ + . . . + aj^_2) = 

therein one may put the coefficient Sq = 1, without impairing the gen- 
erality; the designation bq will, however, be retained. For all 
remaining coefficients a2_ to an_2^ ^° additional restriction will be 

prescribed other than that they are to be real and not all zero (which 
would be trivial for this statement of the problem). Multiplication of 
the two factors yields 



aQ^^^ + a-LX,""-*- + (ag + aaQ)x^"^ + (a^ + aa^^)^'^"^ + 
Aq Ai Ag Ao 




The combined coefficients A one now visualizes as substituted in 
the Hurwitz determinant Dn-i? "the general term of which has in the 



NACA TM 13^ 



sth column and zth line the subscript x = 2s - z. Thus, one has there 
Aps-z °-'^> expressed by the coefficients a, the term 



32S-Z + aa2s-z-2 



The value of the determinant Dn-l is now extended by multiplying 

all terms of the first line by Bq and all terms of the second line 

by a-]_. Fiirthermore, one maies use of the theorem that a determinant 

does not change its value if one adds to the elements of one line the 
elements of another line multiplied by an arbitrary number. This is done 
by adding to the 'terms of the first line, which already have been multi- 
plied by a„, the terms of the third, fifth, . . . line multiplied by 

ap, a. , . . .j thus, one obtains as the general term of the sth col\amn 

in the first line 



3z-l(a2s-z + a32s_2_2) 
Z— J_, J, p, • • • 



Likewise, one adds to the terms of the second line already multiplied 
by a-|^ the terms of the fourth, sixth, . . . line multiplied by a-D, 

a^, . . .; one then finds as the general term of the sth column in the 
second line 



, , ^z-l(32s-z + aa2s-z-2) 
z=2,4,6,... 



In order to calculate these sums, it is sufficient to note that one 
must generally put a^ = when the subscript x is negative; there 

then results for a term of the first line 



NACA TM 13^ 



Z_ az-l(a2s-z + 3a2s-z-2) = aoa2s-l + 32^23-3 + • • • + 



^2s 



_l^a3 + ags.pai + a(ao32s-3 + ^2^23-5 + • • • + 323-4^1) 



and for a term of the second line 



9z-l(a2s-z + 9^23-2-2) = 31323-2 + 33''23-if 
z=2,i^,6,... 



^2s-3''2 + ^23-1^0 ^ ^(^1^2s-4 ^ ^3^23-6 + • • • + 32g_3aQ) 



One can 3ee that both 3ums are of equal magnitude which, however, 
does not signify anything else but that the corresponding elements of 
the first two lines are equal and that therefore the Hurv/itz-determinant 
^n-1 extended by aQa-]_ identically disappears: 



^O^lVl = ° 



From the derivation follows that one could have extended initially, 
instead of the first and second line, two arbitrary other odd and even 
lines by ag_-j_ and then have proceeded further in the same manner; one 

then would have obtained quite generally 



a a -^D , = 
even odd n-1 



8 MCA TM I3W 



Under the obvious assumption that there always exists such a pair 
of values of the coefficients a^ to an-2' ^^^ product of which does 
not disappear, thus 



Dj^_-L = 0, Q.E.D. 



is valid. 



3. From the derivation of this proof, one may f\irther conclude that 
part of the curve Dji_i = may run entirely in the unstable region; 

thus it does not appear there as the required stability limit. Since 
such a possibility had been pointed out before, the motivation of this 
noteworthy indication should be mentioned. 

In splitting the prescribed equation of the nth degree into two 
factors, no more detailed data on the coefficients bq to 82 °^ 

the expression of the (n - 2)th degree have been given and accordingly 
none regarding the roots of the equation 



,n-2 n-1 



...) 



either. 



These roots may therefore have negative as well as positive real 
parts in the neighborhood of points of the plane of representation for 
which Dn-1 disappears; in other words, they may signify stability and 
also instability. If there exists at least one root of the former 
expression of the (n - 2)th degree with a positive real part, this fact 
signifies that the curve E^.j = lies in an unstable region. 

Figure 3 is to indicate how, for instance, three roots may vary 
their position in the complex number plane if their point of reference 
in the plane of representation shifts beyond the curve D^-l ~ ^' Thus 
Dj^_j_ = is not necessarily always the boundary between two regions 
where stability and instability prevail. 

Neither does the sign of the coefficient a in the quadratic expres- 
sion ( X,^ + a) play a role in the derivation of the proof. If a is 



NACA TM 13^ 



positive, a purely imaginary pair of roots satisfies the equation 
( A,2 + a ) =0; this case has been considered in particular. Hov/ever, 
a could just as well be negative without altering the result of the 
theorem. This signifies, however, that in the presence of two real 
opposite-equal roots of the equation of the nth degree, the Hurwitz- 
determinant of the (n - l)th degree also disappears. 

As figure k shows, here also an unstable region lies to both sides 
of the curve D^-i = so that again it does not appear as the sta- 
bility limit. 

Thus, these examples show that it is indispensable to make sure 
whether really all stability conditions on one side of the curve 
Dj^_-[_ = have been fulfilled; only then that curve forms together with 

the curve aj^ = the boundary against the unstable region. 



Translated by Mary L. Mahler 
National Advisory Committee 
for Aeronautics 



REFERENCES 



1. Price, H. L. : The Lateral Stability of Aeroplanes. Aircr. Engng.. 

Vol. 15 (19^3), No. 173, 17^. 

2. Hurwitz, A.: Ueber die Bedingungen, unter welchen eine Gleichung 

nur Wurzeln mit negativen reellen Teilen besitzt. Mathematische 
Annalen XLVI. 



10 



NACA TM I3I18 



ii Dynamically unstable 




Dn-l=0 



o Point at which 
stability prevails 




an=0 



Statically unstable 



Figure 1.- Travel of the point after establishment of stability. 



NACA TM 13i;8 



11 



R = ' 



E>0,R>0 

Instability 
possible _ 



Dynamically^ 
unstable 





R=0 



Stable 



..^^rgr^?;^^^^^ 



E = 



^^r* 



Statically 
unstable 



Figure 2.- Graphic representation of the region E > 0. 



12 



NACA TM 13i+8 



'n-1 



NUnstableV^lv 



-4t. 



■-4r 



NUnstobleOC^b:; 

^ ^ ^ W \ s\ 



-^ 

^ 



Figure 3.- Change of position of the roots in the complex number plane. 



1 = 



W N NN \N N"'C 






^ 



Nonstable W^_ ^^ ^'-^p 



\;Unstable C^-t^ ©Xx 

^ ^ \ \ \ \ 



Figure 4.- Unstable region for the curve 



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